2.2. LORENTZ INVARIANCE 55
or equivalently,
H 1 =
∂A 3
∂x^2−
∂A 2
∂x^3, H 2 =
∂A 1
∂x^3−
∂A 3
∂x^1, H 3 =
∂A 2
∂x^1−
∂A 1
∂x^2,
E 1 =
∂A 0
∂x^1−
∂A 1
∂x^0, E 2 =
∂A 0
∂x^2−
∂A 2
∂x^0, E 3 =
∂A 0
∂x^3−
∂A 3
∂x^0,
the first pair of the Maxwell equations (2.2.33) and (2.2.34) are in the form:
(2.2.39)
∂Gμ ν
∂xν= 0 forμ= 0 , 1 , 2 , 3.The second pair of the Maxwell equations (2.2.35) and (2.2.36) are
(2.2.40)
∂Fμ ν
∂xν=
4 π
cJμ forμ= 0 , 1 , 2 , 3.It is clear that the Maxwell equations (2.2.39) and (2.2.40) are covariant under the Lorentz
transformations. In fact, (2.2.39) and (2.2.40) can be equivalently written as
∂νGμ ν= 0 , ∂νFμ ν=
4 π
cJμ.Also, two electromagnetic dynamic equations and the chargeconservation law are written
as
(2.2.41)
∂ ρ
∂t+divJ= 0.The motion equation in an electromagnetic field is given by
(2.2.42) m
dv
dt=eE+e
cv×H.It is clear that (2.2.41) can be written as
(2.2.43) ∂μJμ= 0 ,
which is Lorentz invariant.
The Lorentz covariance of the motion equation in an electromagnetic field follows from
the following Lorentz covariant formulation of (2.2.42):
(2.2.44) m
duμ
ds=
e
c^2
Fμ νuν forμ= 0 , 1 , 2 , 3 ,whereuμis the 4-D velocity given by (2.2.26), andFμ νis as in (2.2.38).
In summary, all electromagnetic equations can be written inthe Lorentz covariant forms
given by (2.2.39)-(2.2.40), (2.2.43) and (2.2.44), and the Lorentz invariance of these equa-
tions follows.